CONDUCTION BAND ANISOTROPY ON THE 1/1 EXCITON … · 2014. 9. 27. · 2nd USAF Lt Graduate Engineering Physics December 1982 ... would never have been completed. Captain George Norris

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EFFECTS OF CONDUCTION BAND ANISOTRO;"YON THE EXCITON-PLASMA MOTT TRANSITION

IN INDIRECT GAP SEMICONDUCTORS

THES IS

AFIT/GEP/PH/82D-7 Barry S. Davies- 2nd Lt USA" ~ L iT )

FEB 2 21983

App)roved for public release; distribution unlimited

. .- . . - -.. * - I.... ---..--.- I---*.

APIT/GEP/PH/8 2D-7

EFFECTS OF CONDUCTION BAND ANISOTROPYON THE EXCITON-PLASMA MOTT TRANSITION

IN INDIRECT GA~P SEMICONDUCTORS

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

in Partial Fulfillment of the

Requirements for the Degree of

* - .Master of Science

i-.- -

byn

2nd Lt USAF

Graduate Engineering Physics

December 1982

Approved for public release; distribution unlimited

Acknowledments

Special thanks and credit should go to several people

without whose participation this project would never havebeen completed.

Captain George Norris provided the day-to-day supervi-

. sion. His knowledge of both the big-picture and the fine

details of the project, and his willingness and ability to

impart that knowledge, enabled me to accomplish far more

than would have been possible otherwise.

Dr. K. K. Bajaj approved the project initially, pro-

vided several helpful discussions along the way, and made

essential suggestions for the oral presentation.

Dr. Shankland contributed computer routines for numeri-

cal integration and function minimization. Also, my dis-

cussions with Dr. Shankland made the numerical analysis and

programming problems easy where they would have been quite

difficult otherwise.

Captain Dwight Phelps provided the key suggestion for

finding the value of a non-tabulated integral.

Finally, special thanks should go to Jill Rueger for

the typing of the final draft. Certainly there are few

people who could do so much, so fast, so late.

. . .

. ' - "- ' . . .' " . . . : " ' •- "'" o,. ... .. .,,*4. ..-. I . . -. . o. . 4 .

Contents

Acknowledments . ..

List of Figures..... ........ .. . iv

List of Tables . . . . . . . . . . . . . . . .. . .. v, Abstract . . . . . . . . . . . . . . . . . . . . . . . vi

A . Introduction . . . . . . . . . . . . . . . . . .Exciton Formation ..... . ... .. 2

The Exciton-Plasma Mott Transition ..... 3

II. Experiment . . . . . . . . . . . .5

Luminescence Spectra. 9999.... 999 7The Threshold Pumping Powers . . . . . . . . 10The Phase Diagram ............. 13The Theoretical Problem . . . . . . . . . 14

SIII. Theory ..................... 1

Background . . . . . . . . . . . . . . . . . 15The Exciton Hamiltonian . . . . . . . . . . 18The Variational Calculation . . . ... . . . 24

The Kinetic Energy: . . . . . . . . 25-.: The Potential Energy: ...... • 26

Minimization of the Total Energy . . . 32: IV. Results ..................... 34

V. Summary and Conclusions . ............ 39

Bibliography . . . . . . . . . ............ 40

Appendix: The Anisotropic Dielectric Function . . . . 42

ii

..........................................

* .* List of Figures

Figure Page

1 Exciton Formation. * .. .. . 3

2 Phase Diagram... .. . .. . 6

3 Luminescence Spectra (180K) . . * . . . . . . 8

4 Luminescence Spectra (236K) * . . % * . e . . 9

5 Luminescence Spectra (300K) . . . 0 . . . . . 11

6 Determination of Threshold Powers . . . . . . 12

4iv

List of Tables

Tab1- Page

I Material Paree. . . .. . .. .. ... 35

II Mott TransitinSi.. .. .. . .. . .. 36

III Mott Transiiin . .. . .. . .. .. 37

IV Binding Energies for Zero Screening . . . . . 38

'.Abstract

A theory is developed for the incorporation of conduc-

tion band anisotropy into the analysis of the exciton-plasma

Mott transition in indirect gap semiconductors. Ellipsoidal

energy surfaces are assunpd for the electrons while spheri-

cal energy surfaces are retained for holes. Static electron-

hole screening in the random phase approximation is assumed.

The Mott transition is associated with the electron-hole

-pair density at which the exciton binding eneray in the

assumed potential is zero. The binding energy is computed

variationally.

It is found that the electron anisotropy causes the

Mott transition to shift to higher densities. It is also

W -found that, in the absence of screening, the exciton binding-energy is not significantly affected by the electron aniso-

- .tropy. It is thus concluded that the shift to higher densi-

ties is due largely to the reduced ability of anisotropic

-electrons to screen.

Vi.

U --| -

* EFFECTS OF CONDUCTION BAND ANISOTROPY ON THE EXCITON-PLASMAMOTT TRANSITION IN INDIRECT GAP SEMICONDUCTORS

I. Introduction

The exciton-plasma Mott transition in silicon has been

studied in detail in receht years. The experimental data

(Forchel, 1982; Forchel, to be published) has been success-

fully explained (Norris and Bajaj, 1982) by assuming that

the electron-hole interaction is statically screened, and by

taking the conduction and valence bands to be isotropic.

While the assumption of conduction band isotropy works well

in Si, where the ratio of longitudinal to transverse electron

masses is less than five, one would expect the effects of

* conduction band anisotropy to be larger in Ge, where the ratio

of longitudinal to transverse electron masses is greater than

nineteen. It is thus the purpose of this work to extend the

previously mentioned theory by taking conduction band ani-

sotropy into account.

The system under study here consists of optically

generated electron-hole (E-H) pairs in pure indirect gap

"" semiconductors. The presentation will therefore begin with

a discussion of how excitons are formed through optical

pumping, followed by the definition of the exciton-plasma

Mott transition.

In the next chapter, the experimental work will be

discussed. This will be done to show the experimental

.-- p.

evidence for the occurrence of the exciton-plasma Mott

transition, and to show clearly how the connection between

theory and experiment is made.

The theory will be developed in chapter three and

results for both Si and Ge will be presented in chapter

four. Finally, the conclusions will be summarized in

chapter five.

Exciton Formation

The formation of excitons in an indirect gap semicon-

ductor is shown schematically in Figure 1 (Wolfe, 1982). A

photon, which has an energy greater than the energy gap (Eg)

of the semiconductor, is absorbed and excites an electron

from the valence band to a conduction band state which lies

Uabove the conduction band minimum. The electron then under-goes a rapid thermalization process in which it loses energy

to the lattice through the emission of phonons and thus

relaxes to the conduction band minimum. Hydrogen-like

*bound states exist within the energy gap. They are due to

bound E-H pairs, which are called excitons, and which may

form if the sample is sufficiently cool (e.g. T < 80K for

Si). The electron and hole eventually recombine giving off

a characteristic luminescence. Since the material is in-

direct gap, the E-H recombination is also accompanied by

the emission or absorption of phonons.

2

+ .4

4J

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P4

r4

00

41)0

c

e Xe

NO

"2-I

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r4

* 0

x

r-4Am0'K_4

~S1

The Exciton-Plasma Mott Transition

Consider a free exciton (FE) gas and ask the question,

"How can the bound electrons and holes become dissociated?"

One way in which the E-H pairs may become dissociated

is through thermal ionization. This is a diffuse process in

that, for a given temperature, the FE gas and EHP coexist in

thermal equilibrium and there is not a precisely defined E-H

pair density at which dissociation occurs.

Another way in which E-H pairs may become dissociated

is through entropy ionization (see for example: Mock, 1978).

This is a complicated effect which occurs when the total

number of electrons and holes in the system is reduced at.

constant temperature. The important point here is again

that there is no precisely defined E-H pair density at which

dissociation cccurs.

The third way in which excitons may ionize is through

screening. In this case the exciton density is increased,

at constant temperature, by increasing the optical pumping

power and a point is reached at which a given electron can

no longer be associated with any particular hole. Here there

is a precisely defined density at which the exitonic binding

energy goes to zero. The transition is from an insulating

FE gas to a metallic electron-hole plasma (EHP) and is called

a Mott transition.

Thus, the Mott transition is an insulator-to-metal

transition which arises as a result of screening.

4

II. Experiment

Experimentally, the exciton-plasma Mott transition has

been studied in connection with a separate transition which

is known to occur and which is well understood. This other

transition is a first-order phase transition in which the

FE gas condenses into a highly dense metallic electron-hole

liquid (EHL). Experimental work may be seen for example in

Hammond et al (Hammond, 1976), Thomas et al (Thomas, 1973),

and Thomas et al (Thomas, 1974). A thorough discussion of

the theory of the EHL is given by Rice (Rice, 1977).

The EHL is important in the experimental study of the

Mott transition and will thus be briefly discussed.

The phase diagram for the electron-hole system is shown

in Figure 2. How such a phase diagram is constructed will

be discussed shortly. For the present it is sufficient to

note that below a certain critical temperature, Tc, a two-

phase region exists where the FE gas (or EHP) is in equili-

brium with the dense EHL. The solid curve is the liquid-gas

coexistence curve. If the FE gas density is increased (by

increased optical pumping at sufficiently low, constant

temperature), the gas-to-liquid transition will occur when

the FE gas density reaches the value on the coexistence

curve. The EHL will then be present in droplets whose den-

sity is determined by the liquid side of the coexistence

* curve. Further increases in optical pumping change the size

. -.7 of the electron-hole droplets but do not change their density.

. .

707

70I

600

50 Exciton Electron-HoleGas Plasma

40

3 0

4

20

10 Two-Phase Region

i1 101 101 no 10l

Density (cm)

Figure 2: Phase Diagram

Given the above comments on the EHL, one may now turn

to the experimental construction of the phase diagram of

Figure 2. This construction will be illustrated through a

discussion of a specific experimental study of the exciton-

plasma Mott transition (Shah, 1977). Shah and his co-workers

were first to report experimental evidence for the Mott dis-

sociation of excitons into EHP. Their work shows clearly

how the experimental data are obtained and interpreted. This

discussion will thus serve to define the theoretical problem

as well as to provide the experimental background.

The Luminescence Spectra

Shah and his co-workers excited a crystal of pure Si

with radiation from an argon laser and observed the resulting

WY luminescence spectra at various temperatures and for variousoptical pumping powers.

The spectra which they observed at low temperature (18°K)

are shown in Figure 3. For low pumping powers, the FE line

was seen, while for sufficiently high pumping powers, a second

.- peak due to EHL luminescence was also present. Both of these

luminescence peaks were found to have line shapes which were

independent of pumping power.

Figure 4 shows the spectra which were obtained at 23*K.

For low pumping power, the FE line was again seen. However,

as the incident power was increased, the low energy side of

the FE line was observed to broaden as shown by the singly

[ . dashed curve. Again, for sufficiently high pumping powers,

7

" .-

ri4

04

'4)

r14)

sousosulu0

qq

'44

Im

r4.

u Ul OU90OUTA

the second peak due to the formation of EHL was observed.

The spectra obtained at 30*K, which is above the criti-

cal temperature for the formation of an EHL, are shown in

Figure 5. At low incident power the FE line was observed

(curve 1), and as the power was increased, the line broaden-

ing was seen to evolve continuously into a shape which is

well fit by a plasma line shape (curve 3).

Shah and his co-workers interpreted the observed line

broadening as being due to the Mott dissociation of excitons

into an EHP. This is a logical interpretation since the FE

line shape is independent of pumping power, while the line

shape due to the recombination of unbound electrons and holes

does depend on pumping power. It is thus natural to associate

the onset of the line broadening with the onset of the Mott

dissociation. How this was quantified will now be presented.

The Threshold Pumping Powers

In order to determine the threshold powers for the Mott

transition, and for the formation of EHL, Shah and his co-

workers plotted the change in position of the low energy

half-maximum of each distinct peak observed in the spectra.

The change was plotted as a function of average incident

power. The results are shown in Figure 6.

At 18°K, only the FE and EHL peaks were seen and their

shapes were independent of pumping power. Thus the power,

IT at which the EHL peak was first observed was taken as

the threshold for EHL formation.

10

00

CllM

94H

V

02Mo

Hnt

N r4

ATGU8UI 9u83su~uK

FE

2T

4-4

(d 4JFE

10 25 K

11I$4 I

~94 EHL

4'~ 3

0

44

FE

1 Plasma 30 K

3

0.1 1 10

Average Incident Power (WO

-Figure 6: Determination of Threshold Powers (Shah, 1977)

12

; -T T7 7 7.

At 25"K, the line broadening was found to yield a linear

plot between the FE and EHL plots. IT was determined as

before for EHL condensation. The threshold, ID' for the

onset of line broadening was determined by extrapolation as

shown.

At 30K, the continuous broadening was observed and ID

was again determined by extrapolation.

Once the threshold powers were determined, there

remained the problem of converting those powers into E-H

pair densities. This problem was solved by constructing the

phase diagram for the E-H system.

The Phase Diagram

The phase diagram for the E-H system in Si was shown in

Figure 2. The solid curve is taken from Norris and Bajaj

(Norris, 1982), while the circle and triangle points are

included to aid in the description of how Shah et al con-

structed a similar phase diagram.

The EHL densities (triangular points) were determined

* from the liquid luminescence half-width by using the theo-

retical calculations of Hammond, McGill, and Mayer (Hammond,

1976). This process resulted in the experimental determina-

tion of no , the EHL density at OK.

Given experimental values for nO and Tc (the critical

temperature for EHL condensation) the theoretical calcula-

tions of Reinecke and Ying (Reinecke, 1975) were used to

obtain the liquid-gas coexistence curve (solid line). It

13

was thus possible to determine the FE gas densities (circles

on the solid curve) which were in equilibrium with the EHL

at various temperatures, and to associate these densities

with the measured threshold powers (IT ) for EHL condensation.

The Mott transition densities were then determined from

the threshold powers, ID,' by assuming a temperature-indepen-

dent, linear scaling between E-H pair density and pumping

power. This assumption is of uncertain validity but repre-

sents the best method available for the experimental deter-

mination of the Mott transition densities.

The Theoretical Problem

The above discussion clearly defines the theoretical

problem for the Mott transition: One must predict the densi-

ties at which excitons become unbound due to screening. A

theory for doing this will be discussed in the following

chapter and will be extended to take conduction band anisot-

ropy into account.

14

............ .. .. .. .. .... . . . . . ..- ......

III. Theory

As was mentioned in the introduction, Norris and Bajaj

(Norris, 1982) have developed a theory for the exciton-

plasma Mott transition in Si which is in good agreement

with experiment. They obtained the exciton Hamiltonian by

assuming isotropic masses and by assuming static electron-

hole (E-H) screening in the random phase approximation

(RPA). They then associated the Mott transition at a given

temperature with the E-H pair density for which the binding

energy of the exciton becomes zero. The binding energy was

evaluated variationally. In this chapter, the above theory

will be presented and extended to take into account the con-

duction band anisotropy. As was mentioned earlier, this is

being done in order to assess the effects of the electron

anisotropy on the exciton-plasma Mott transition both in

Si and (more importantly) in Ge.

Background

Theoretical work on the exciton-plasma Mott transition

was preceded by investigations of the similar problem of an

electron bound to a donor impurity in a many-valley semi-

conductor. In the latter case, the task is to compute the

donor concentration, Nc , at which the electrons become

unbound due to screening. Since the theory to be presented

in this work is a direct product of the earlier work, the

donor impurity problem will now be reviewed.

15

• .. - - - .' . . " - . . . . . "-. - - . . .

%;-:1

Initially, the screening was taken into account by

assuming the electron to be bound in a Yukawa potential,

and the variational calculations were done using hydrogenic

trial functions (Mott, 1949). Later, Rogers et al (Rogers,

1970) numerically integrated Schrdinger's equation for

the above case. The two results were not in very good

agreement, and hydrogenic wave functions were seen to be

poor trial functions.

Later, Lam and Varshni (Lam, 1971) did the variatio.al

calculation for the Yukawa potential using eigenfunctions

of the Hulth~n potential. Their results were in good agree-

ment with the calculation of Rogers. The Hulth~n potential

and its s-state eigenfunction, as given by Greene et al

(Greene, 1977) are

: e 2ije-PrV I(r) -e-Pr)

and

.(r) =(/2)r/a - + p/2)r/a(.- '" -- 2(wa) r

where e is the electronic charge, £o is the static dielectric

constant, a is the first Bohr radius of the electron, and

is taken as a variational parameter. Greene et al used the

Hulth6n wave function to solve the problem of an electron

in the Lindhard potential (RPA) at T = 0 and of an electron

in the Hubbard-Sham potential (which includes first order

16

! ° . . .. .-.. . ... -... .. .. . °.

,,.,:,', S ,. , S ,,.........-, .',....... . .... , .. ... . . . . . ,..." • . . S

I.7-

corrections to the RPA at T = 0). Their result for the

Hubbard-Sham potential was in good agreement with the result

of Martino et al (Martino, 1973), who solved the corresponding

Schr~dinger equation numerically. Thus the Hulth~n wave

function was again seen to be a good trial function.

In all of the above *ork, the electron masses were

taken to be isotropic. Since the electron masses in many-

valley semiconductors are anisotropic, Aldrich (Aldrich, 1977)

treated the problem of an electron with anisotropic mass

bound in the Lindhard and Hubbard-Sham potentials at T = 0.

He performed the variational calculation using a modified

form of the Hulth~n wave function, which will be used in

the present work:

(r) = ap (j/ 4- /a p P/2a - e-P/ 2a).(3/ 4-p e e/ (3)

where a is the Bohr radius, $ and v are variational para-

meters, and where P is given by:

p F a x + at(Y2 + z 2 (4)

2 m 2

Ellipsoidal energy surfaces are assumed so that longitudinal

and transverse masses, me and mt, may be introduced, and': * -(m2 )1/3

(mt-) . The parameters at and at are given by

'"' B 2/3a£ = (5a)

17

,, ..:.-: . .. * .: .-. .. . - . . - . . -, . . . .- .. . . . . . . : . : : . - . . : -

*."-.a =- .. / 1 / 3

a O, (5b)

where c is a third variational parameter. The variational

parameter, V, reflects the strength of the Hulthdn potential,

B effectively expands or contracts the wave function, and c

adjusts the trial functioA anisotropy.

The above discussion shows how the choice of RPA

screening has come about and how the variational approach

has developed. The theory of Norris and Bajaj for the

exciton-plasma Mott transition will now be presented in

*" detail via its extension to include electron anisotropy.

This will be done by considering first the exciton Hamil-

tonian, and then the variational calculation for the excitonic

binding energy.

The Exciton Hamiltonian

The exciton Hamiltonian is taken to be

+=42 3[ 1 2 1 92] U(-h (6': H + (6

ei

where 1/mei and 1/mhi (i = 1,2,3) are the diagonal elements

of the electron and hole effective mass tensors. The posi-

tion vectors for electrons and holes are Ee and Eh, while

Xei and xhi are the canponents of these vectors. U(e-rh)

is the interaction potential.

In writing the kinetic energy term for the electron

in equation (6), ellipsoidal energy surfaces have been

.'- 18

assumed. The hole mass will be taken to be isotropic: The

effective mass tensor notation has been retained in the

kinetic energy term for holes in order to make the deriva-

tions easier. The assumption of isotropic hole masses has

been made because any attempt to go beyond this assumption

would make the problem intractable. For a complete treat-

ment of the hole kinetic energy see (Lipari, 1971), and for

a discussion of semiconductor band structure see (Rice,

1977: page 5).

A center of mass transformation is now made in order

to treat the e-h pair mathematically as a single particle

in the assumed potential. Thus

r -e -rh (7a)

Xi 'neiei + mhiXhi (7b)mei + mhi

I _ + 1_ (7c)i mei mhi

where the mi are reduced masses. Ignoring the translational

term, which does not affect the binding energy, one obtains

2 3 i 2

H=- E U(r)(8

Since the energy surfaces are assumed ellipsoidal,

longitudinal and transverse effective masses may be

introduced:

19

met = mel , met = me2 = me3 (9a)

.ht = mhl ,mht = mh2 = mh3 (9b)

It should again be noted that the hole masses are assumed

to be isotropic and that the introduction of mht and mht

is solely for ease of derivation.

The reduced masses are thus given by

1 1 1m- = -- (10a).::. L et mlt

1 + 1 (10b)mt met mht

and the exciton Hamiltonian is given by

+ + ;U(r) (11)

In order to complete the determination of the Hamil-

tonian, an explicit expression for the potential energy must

be obtained. This is done in wave-vector space (the Fourier

transform domain of position space) by assuming static

electron-hole screening in the random phase approximation.

It is thus assumed that the electrons and holes respond to

the unscreened (Coulomb) potential individually rather than

collectively. It is thus also assumed that the total

potential can be written as the unscreened potential plus

20

L:': "- "'-". '. '".,_ . • . : , ."- " ,"." ' ", . " .. . " ,-"- .'". " , "_ ; --: ;: £ : ._:, L , 2 :, . .- ..- - ' .... . .:,: " ."::".i

a term which represents the average response of the screen-

ing carriers to the total potential. For electron screening

this is

r 3V(r) =V (r) d r (12)0 -J;:r4

where V(r) is the total potential at r, VoCr) is the

*~' unscreened potential at r, and is the expectedr

value of the screening particle density at r'.

The particle density is given by

Ar = Trace(p [Vln(r)) (13)

where pIV] is the density matrix and n is the particle

density operator. The density matrix is obtained from its

equation of motion

ih P= [1PHI (14)at

. where H is the system Hamiltonian. The density matrix and

-Samiltonian are written in terms of perturbations:

P 0 + SO (15a)0

.H=H + V (15b)

The subscript "zero" refers to the problem in the absence

of an external potential and V includes both the external

and screening potentials. The RPA arises when equation (12)

is assumed and the term in 8pV is dropped from equation (14).

21

When the above analysis is carried out for both

electrons and holes, the potential energy is found to be

given by

4 2 1V () T ne (16)

r a

where e is the electronic charge, eo is the static dielectric

constant, q is the wave-vector (of magnitude q), and where

*c(q) is the dielectric function. The dielectric function

is given by

C(S) = 1 - 47re g + vhgh(q)] (17)Coq

where v e and vh are the band degeneracy factors for electrons

and holes, and where the g functions are the density-density

response functions for electrons and holes. The response

function for electrons is

Id3k f[(Ee(k)-e)/kbT]-f[ (Re(k-q)-pe)/kbT]ge ( ) = 8 ' e()-Ee (k-)

vith a similar expression for holes. In equation (18), d3k

is the wave-vector volume element dkxdkydkz, f is the Fermi-

Dirac function f(x) = 1/(ex + 1) E e is the energy as a

function of wave-vector, ie is the chemical potential for

electrons, and kbT is Boltzmann's constant times the absolute

temperature.

22

.o

Now, the integral in equation (18) has been evaluated

for practical computation by Meyer (Meyer, unpublished),

assuming spherical energy surfaces. In order to do the

present work, it was therefore necessary to assume ellip-

soidal energy surfaces according to

E (k) = k2 + k2 (k2 + k 2 (19)TM __ y zEe 2mel = x et

and cast the expression for the dielectric function into a

form where Meyer's results could be used. The details of

this process are outlined in the appendix. The result is

2FF , 14wne (X- () + h)nXe 1L.~ (Xeeoeel + (i h q(20)

where n is the electron-hole pair density, the n's are the

reduced Fermi energies (the chemical potentials divided by

kbT), and where

+hxo 1

G(,Xo)- F-#0 f(x-ntn -dx (21)(lrx 0 )

1 k --

Pk(n) = 1 - ndx (Fermi integral; order k) (22)0

and

23

-. -'.' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...- '.+-• ..'. ., -" --- - '-. A S,... "'.' ..'+-,2 ,,--'-., ,"• .

2e 2 2-4.'..-. _"" me e 2m2e

x m e q2'+ -(q + qj (23)eo m*kbT qx met I' z

with a similar expression for Xho. In equation (23)

Cm2 1/3e emet)/ . Again f is the Fermi-Dirac function.

Meyer has cast the function G(xo,n) into approximate analy-

tical form and his results can (and will) be used in the

present work provided equation (23) is used for Xeo along

.* - with the corresponding expression for xho.

The explicit expression for the potential energy term

in the Hamiltonian has now been obtained (equation 16 com-

bined with equation 20). The variational calculation for

the exciton binding energy will now be presented.

The Variational Calculation

It was mentioned at the beginning of this chapter, that

the Mott transition at a given temperature is associated

with the E-H pair density for which the exciton binding

energy is zero. The binding energy is computed variation-

ally by minimizing the expectation value of the Hamiltonian,

, where is computed using the Hulth~n wave function

given in equation (3). This section will therefore address

the determination of = + , where and

are the expectation values of the kinetic and potential

energies, and will conclude with a discussion of the mini-

mization process.

24

.- . .. .

The Kinetic Energy:

* ." As previously discussed, the trial function for the

variational calculation is

2 3/ 4-2] e-P/a(r) -377 -T ih(4

* where p, is given by

= atx2 + a(y 2 + z2 (25)2t 2 m2 )

and where a = o42 /m e 2 is the exciton Bohr radius. The

" masses mt and mt are the longitudinal and transverse reduced

* masses introduced in the discussion of the Hamiltonian,

while m* . (mtm2) / 3 . Also, at = Oe2/ 3 and at = 8/c I / 3

4 as previously defined. The variational parameters are V, B,

and c.

The expectation value for the kinetic energy,

* (* _h2 [1 a2 (2 + 2 2 'I3r()

. .. . . . -

The Potential Energy:

The potential energy can not be expressed in closed

form in position space and its expectation value is, there-

fore, determined by a calculation in wave-vector space. In

the usual expression for the expectation value for U(r),

= j*U~d3 r (28)

U is expressed as the inverse Fourier transform of the

potential, V(a). Thus

= VP*L.1 L 7 V()eii' d q q d3r (29)

where V(g) is the statically screened Coulomb potential

given by equations (16) and (20).

It will be seen later that the determination of

involves the numerical evaluation of a double integral on

the unit square. It turns out that if the potential, V( },

is used "as is" in equation (29), then the integrand will

have a finite value on the side of the unit square which

corresponds to infinite wave-vector magnitude. This diffi-

culty can be eliminated if V(a) is expressed as the sum of

a "screening" term and a Coulomb term. In this case, the

expectation for the Coulomb term can be evaluated analyti-

cally, while the integration for the screening term involves

an integrand which vanishes on the previously mentioned side

of the unit square.

Thus one writes

26

-. -.. .

41re2Vla) = 4-e2 = Vslql + Vclgl (30)

£oq2C Ql)

where

- S 4re2 c1 - (31)-=~~~ ~~ Vsq - -E-11

oco

and

Vc ---- re2 (32)2

and where c(q) is the dielectric function.

-When equation (30) is substituted into equation (29).

for , two integrals result.

The Coulomb integral is found to be given by

,.. = H h(p) h(e) (33)

where H* is the Hartree, where p, 1, and e are the varia-

tional parameters in the trial wave function, and where the

functions g(p) and h(£) are given by

4 2 ]n (34a)

and

27

. . . . .-

.

'-. . .- . .." ,-.,. - >.. . . . . . -

C:i si- 1 t \Cmth (c)

(~~:) -_ L-mL Yr 1 >(34b)

For the screening term, one has

1* Vs (a) eiS'Td3q d3r (35)

NOW, can be expressed as a single triple integration

in q-space by making the change of variables from q to -

and interchanging the order of integration. One then obtains

the integral over all q-space of Vs (a) times the Fourier

transform of the wave function squared:

)V(a)d q (36)(2wr)7

The Fourier transform, F, is easy to compute and is

given by

tn12 laa + tanq' 2 tan-l(a

where

28

:9 ' . - ,.-. ° . .. ". - .. '. -- - . . . .. .. -. °. .- . . ... .....-........ . .- . .. . .

q' - qX + m t (qy + q) (37b)

The masses which appear in equation (37b) are the reduced

masses, a is the Bohr radius, and the quantities at and

a were defined in the section where the wave function was

introduced (see equations 3-5).

The integral in equation (36) is evaluated after several

changes of variables.

First, the spherical coordinates (q, 8, ) defined below

are introduced.

qx = qcosO (38a)

qy = qsin~cos* (38b)

q = qsinesin (38c)

The integration with respect to * can be done immediately,and one is left with a double integral. However, because of

the complicated nature of the integrand, neither the q- nor

the e-integration can be performed analytically and the

double integral must be computed numerically.

The second and third changes of variables in equation

(36) are to introduce respectively q = aq and u = cosO

Again, a is the Bohr radius, and the "tilde" is used on q

to indicate that is a dimensionless wave-vector magnitude.

29

Thus, if the expression for Vsla) in terms of the di-

electric function (see equation (31) is substituted into

equation (36), one obtains

> 2 *1 ( 3(ui)- - Hf duf dq sF (39

r one has

(ufq-) 4 ( 1) 1 t.-l( + tn..- 2tanl(S;)~

(43a)

with

I (mtl1/6 t 21(4b0- Et 1 - 1- u

and

-C 3(u,4) = 1 + 4(lna3k H 4F ne) G(xeoifne)

+ G G(xho'71 h)~. (44a)F Y nh I q-1

with

m*H* met 1/3 1 144b)et met

and

h.o* H* = '[ (t 2 j12 (4c

VTh

As introduced in equation (22), the fractionally-sub-

scripted F functions are Fermi integrals, and the G function

is the integral which is evaluated by Meyer's interpolation.

It should be noted that all quantities in equations

(43) and (44) have been put into forms where units cancel.

31

3The quantity (na may be identified as a dimensionless

electron-hole pair density, H/kbT may be defined as a

* dimensionless reciprocal temperature, and the quantities

m* H* m* H*m*c and -e b mh

*" may be defined as dimensionless electron and hole reciprocal

temperatures, respectively.

Minimization of the Total Energy

The routine used to find the binding energy, ,

(Shankland, private communication) minimizes a function of

* a given number of parameters. In this case, is a func-

tion of p, , and c, the parameters in the trial wave func-

tion introduced in Chapter III. The search for the minimum

involves random, gradient, average, and jump steps, which

are made initially with user defined frequencies, and finally,

with frequencies generated in the course of the calculation.

. As convergence is obtained, the step size is automatically

reduced to insure that the minimum is found. Finally, jump

steps are made to allow for the possibility of local minima.

There are two possible approaches which can be taken to

find the Mott transition density, nMott. One is to keep the

electron-hole pair density as an input parameter and minimize

as a function of v, 0, and c. The density is then

* adjusted manually until that density is found for which

minimized = 0 . The other approach is to include the

32

density as a variational parameter in the minimization

routine and minimize 2. The first approach has been

adopted here because it is then easier to monitor the course

of the calculation.

The procedure is to make a series of short calls (25

function evaluations each) to the minimization routine,

looking for negative values of . Since the goal is to

find the density for which minimized is zero, and since

the Mott tiansition is due to screening, one knows, as soon

as a negative energy is obtained, that the density must be

increased. The short calls, however, do not allow for the

search step size to be reduced below about 10-1 so that this

initial search is a rough one. Once a series of positive

energies is obtained, than a long call (250 function evalua-

tions) is made to check for the best minimum. In this way

the step size is reduced to about 10-4.

Results will now be presented for Si and Ge.

33

1 17

IV. Results

The material parameters used in this work are given in

Table I (See Rice, 1977: page 8). Masses are given in units

of the electron free-space mass. The longitudinal and

transverse electron masses are given for Si and Ge. The

electron masses used in the isotropic approximation are

* -. obtained from the optical mass average

m 3 ( + (45),..me me t

while the hole masses, mht and mht, are the reciprocal of

the Dresselhaus-Kip-Kittel A parameter given by Rice. The

density of states masses, medos and mhdos, are used in the

qcalculation for the reduced Fermi energies and are deter-

-mined from the following equations:

!"2 1/3(4amedos (metmet) (46a)

g3/2 +i 3 / 2 2/3'-'." mhd (46b)

-:.dos 2

The equation for the hole density of states mass involves

the heavy and light hole masses, mhH and mhL, and thus7.o

--takes into account the fact that the heavy and light hole

bands have been replaced by a single doubly-degenerate band.

Table I also gives the static dielectric constant (c 0 and

the degeneracy factors (veVh) for Si and Ge.

34

Table I: Material Parameters

Si Si (isotropic) Ge Ge (isotropic)

met .9163 .2588 1.58 .1199

met .1905 .2588 0.082 .1199t.2336 .2336 .07474 .07474

Tmht .2336 .2336 .07474 .07474

2edos .3216 .3216 .2198 .2198

Smhdos .3637 .3637 .2247 .2247

11.4 11.4 15.36 15.36

Ve 6 6 4 4

vh 2 2 2 2

The results obtained for the Mott transition in Si are

shown in Table II. The Mott transition densities are given

• as a function of temperature with lower and upper bounds

specified. Thus, for example, at T = 30*K

7.8 x 1016 cm- 3 < nMott < 8.0 x 1016cm-3 -- anisotropic Si

7.4 x 10 6cm 3 < nMott < 7.5 x 1016cm-3 -- isotropic Si

The results of Table I for isotropic Si are in good agree-

ment with the results of Norris and Bajaj (Norris, 1982).

Only two points have been determined for anisotropic Si

35.4

.. . .. . . . . . . . . . . . . . . . . . . .

Table II: Mott Transition in Si

DENSITY (x101 6 cm-3}

Si - ISOTROPIC Si - ANISOTROPIC

T(OK) Lower Bound Upper Bound Lower Bound Upper Bound

10- 4 3.7 3.8

7.81 4.0 4.1 4.1 4.3

15.6 5.1 5.2

23.4 6.3 6.4

30.0 7.4 7.5 7.8 8.0

46.9 10.1 10.2

62.5 12.7 12.8

78.1 15.2 15.3

since two points are sufficient to show the effects of con-duction band anisotropy. It is seen that the Mott transi-

tion shifts to higher densities. The shift is 4% at

T = 7.81*K and 6% at T = 30.0*K. These results will be

interpreted after the results for Ge are presented.

The results for Ge are shown in Table III. Again, the

Mott transition shifts to higher densities when conduction

band anisotropy is taken into account. Here there is no

apparent shift at 5*K while the shift at the higher tem-

-peratures is approximately 7%.

The observations which are to be made here are as

follows: (1) The electron anisotropy causes the Mott

36

9J

Table III. Mott Transition in Ge

DENSITY (xlO 16cm -3 )

Ge- ISOTROPIC Ge- ANISOTROPIC

T(*K) Lower Bound Upper Bound Lower Bound Upper Bound

5.0 .13 .15 .13 .15I

12.5 .26 .27 .28 .29

20.0 .39 ,395 .41 .43

transition to shift to higher densities. (2) The shift is

greater for high temperatures than for low temperatures.

(3) The shift appears to be no greater in Ge than in Si.

The first observation can be explained by the fact that

the anisotropy will reduce the ability of the electrons to

screen and thus a higher density will be required to screen

out the Coulomb potential. The second observation can be

partially explained by noting that the electrons and holes

have more thermal energy at higher temperatures and energetic

carriers do not screen as effectively as less energetic

carriers. The third observation is partially explained by

a calculation of the exciton binding energy in the absence

of screening.

Table IV shows binding energies in Ge for an electron

bound to a hole and to a donor impurity, in the absence of

screening. It can be seen that the anisotropy does not

significantly affect the exciton binding energy. This is

37

'.4

Table IV: Binding Energies for Zero Screening (Ge)

Exciton Binding Energy (meV)

Isotropic Masses 2.67

Anisotropic Masses 2.73

Donor Impurity

Isotropic Masses 6.91

Anisotropic Masses 9.78

because the reduced masses are dominated by the light masses

and in Ge one has met = .082 and mht = mht = .075 . Thus

the ratio of longitudinal to transverse reduced mass is on

the order of only two for the exciton. For the donor

impurity, the above situation no longer prevails and the

anisotropy significantly affects the binding energy. The

result of 9.78 meV is in good agreement with Faulkner

(Faulkner, 1969) who obtained 9.81 meV for an electron

bound to a donor impurity in Ge.

Thus, the shift of the Mott transition densities is

due mainly to the effects of the anisotropy on the screening

since the anisotropy does not significantly affect the

exciton binding energy.

38

"'. - - - . . . ,. .-. --e ,° I" o- - • . * ,,. , - q u o -8 . - . . . - . 'I

V. Summary and Conclusions14

In this work, conduction band anisotropy has been

incorporated into the theory of the exciton-plasma Mott

*- transition (Norris, 1982). The exciton was treated mathe-Imatically as a particle in a screened Coulomb potential

where static electron-hole screening in the randon phase

approximation was assumed. The Mott transition was asso-

ciated with the electron-hole pair density at which the

"* exciton binding energy in the assumed potential is zero,

and the binding energy was computed variationally using the

ground state eigenfunction of the Hulth~n potential as a

"* trial function.

The results obtained from the above theory lead to the

Sfollowing conclusions:

(1) The conduction band anisotropy does not signifi-

cantly affect the exciton binding energy.

(2) The conduction band anisotropy decreases the

ability of the electrons to screen and thus increases the

Mott transition densities beyond those predicted by RPA

screening with the conduction band taken as isotropic.

(3) The effects of the electron anisotropy are no

*more pronounced in Ge than in Si.

39

Bibliography

Aldrich, C. "Screened Donor Impurities in Many-Valley Semi-conductors with Anisotropic Masses," Physical Review B,16: 2723 (1977).

Faulkner, R. A. "Higher Donor Excited States, for Prolate-Spheroid Conduction Bands: A Reevaluation of Siliconand Germanium," Physical Review, 3: 713 (1969).

Forchel, A., B. Laurich, J. Wagner, and W. Schmid. "System-% . atics of Electron-Hole Liquid Condensation from Studies

of Silicon with Varying Uniaxial Stress," Physical ReviewB, 25: 2730 (1982).

. Forchel, (to be published). Phys. Inst. Teil 4, Universitat,D 7-Stuttgart-80, Pfaffenwaldring 57, Germany.

Greene, R. L., C. Aldrich, and K. K. Bajaj. "Mott Transitionin Many-Valley Semiconductors," Physical Review B, 15:2217 (1977).

Hammond, R. B., T. C. McGill, and J. W. Mayer. "TemperatureDependence of the Electron-Hole-Liquid Luminescence inSi," Physical Review B, 13: 3566 (1976).

ULam and Varshni. "Energies of s-Eigenstates in a StaticScreened Coulomb Potential," Physical Review A, 4: 1875(1971).

Lipari, N. 0. and A. Baldereschi. "Energy Levels of IndirectExcitons in Semiconductors with Degenerate Bands," PhysicalReview B, 15: 2497 (1971).

Martino et al. "Metal to Non-Metal Transition in n-TypeMany-Valley Semiconductors," Physical Review B, .8: 6030(1973).

Meyer, J. R. "Analytic Approximation of the Lindhard Di-electric Constant e(q, = 0) for Arbitrary Degeneracy,"(unpublished) 1977.

-Mock, J. B., G. A. Thomas, and M. Combescot. "EntropyIonization of an Exciton Gas," Solid State Communications,25: 279 (1978).

Mott, N. F. "The Basis of the Electron-Theory of Metals withSpecial Reference to the Transition Metals," Proceedings-of the Physical Society of London, 62: 416 (1949).

40

__i

Norris, G. B. and K. K. Bajaj. "Exciton-Plasma Mott Transi-tion in Si," Physical Review B, 26: (to be published)(1982).

Reinecke and Ying. "Model of Electron-Hole Droplet Conden-sation in Semiconductors," Physical Review Letters, 35:311 (1975).

Rice, T. M. "The Electron-Hole Liquid in Semiconductors:Theoretical Aspects," Solid State Physics: Advances inResearch and Applications, edited by H. Ehrenreich, F.Seitz, and D. Turnbull (Volume 32). New York: AcademicPress, 1977.

Rogers, F. J., H. C. Graboske Jr, and D. J. Harwood. "BoundEigenstates of the Static Screened Coulomb Potential,"Physical Review A, 1: 1577-1586 (1970).

Shah, J., M. Combescot, and A. H. Dayem. "Investigation ofExciton-Plasma Mott Transition in Si," Physical Review

- Letters, 38: 1497-1500 (1977).

Shankland, D., Professor, Department of Physics, Air ForceInstitute of Technology, Wright-Patterson Air Force Base,AFIT/ENP, IWPAFB, OH 45433.

Thomas, G. A., T. G. Phillips, T. M. Rice, and J. C. Hensel."Temperature-Dependent Luminescence from the Electron-Hole Liquid in Ge," Physical Review Letters, 31: 386-389(1973).

.....- , T. M. Rice, and J. C. Hensel. "Liquid-Gas Phase

Diagram of an Electron-Hole Fluid," Physical ReviewLetters, 33: 219-222 (1974).

* Wolfe, J. P. "Thermodynamics of Excitons in Semiconductors,"Physics Today, March 1982.

A4

" 41

4.

4. ]J . ,, / ,, .",, "-. ' ,. ,. . < ."-, ,, . ,; e " . -, ."- "- , ,"-" , - ,. - ". _ . " " .": . " " " ' " "- """

Appendix: The Anisotropic Dielectric Function

The screened Coulomb potential in wave-vector space is

given by

4we2V4(a) w (16)coq2 E(g)

where e is the electronic charge, co is the static dielectric

constant, q is the wave-vector magnitude, and eC 1) is the

Lindhard dielectric function.

-' The dielectric function is given by

eca)e =1 l 2 +(17).F.~~~ 7()- [vegel qI) + vhgh (S)O 7-:. COq

where ve and vh are the degeneracy factors and where ge(q)

* and gh(QI) are the density-density response functions for

* electrons and holes respectively. The response function

for electrons is given by

f~)= (E e(k)-Ije)/kbTl-f[F-e(k-S.) -11)/kbTlIge Ee (k) - Eelk a) k118

In equation (18), Pe is the chemical potential for electrons

and f(x) = l/(ex + 1) , the Fermi-Dirac function. The

function, Ee(k), is defined by

x2 2 In 2 2SEe (k) ym kx + m*(ky +k z (19)

42

-.

*- - - - .. . . . .

where mel and met are the longitudinal and transverse electron

effective masses.

The explicit expression for the dielectric function is

obtained by performing the integration in equation (18), with

an almost identical calculation for holes. The first step is

to make the following chahges of variables in order to make

Eelk) and Ee(k-q) spherically symmetric (Aldrich, 1977: 2724):

k m) (met (A-la)kx --/x q = t

k Imet s t (A-lb)

By ~ y m* y

JMet met

k -±-) s = !e (A-lc)Me %

In equation (A-1), m* is given by m = 2e me ie (metmet)11

With the above changes of variables, it is found that

I2 2Ee (k) 2 *--s Eels) (A-2a)

Me

and

Be(k - = I - t1 2 E es t) (A-2b)- *2m -

43I-i .4 -. - .

Of course, the Eels on the two sides of equation (A-2)

represent different functions, but the arguments will

always be explicitly stated, and no confusion should result.

It is found that d3s = d3k so that g e() becomes

1 d 3 f [ (Ee (s )-ue )/kbT ] 1 d3 sf[Ee (s-t)-ue)/kbT]

e E (sd- Sst E (s)-E (s-t)

(A-3)

Both of the integrals in equation (A-3) can be done in

polar coordinates. For both integrals, the s -axis is chosen

to lie along the vector t. Thus, for the first integral, one

has

E(S - t) = (s2 + - 2stcose) (A-4)e 2m

For the second integral, one makes the substitution

s' = s - t , whereby t is eliminated from the numerator.

Since d3 s = d3 s , the primes may be dropped and one then

obtains

,E s + t) A (s2 + t2 + 2stcose) (A-5)2me

in the denominator of the second integral.

After performing the angular integrations in equation

(A-3), it is found that

44

! ' t

" "' '' " ' ' '' '' " '° '' ' °" '" ' '' '' ' "' '' ' a ' " "-° " " '.' " " " ' " ' "" ' .

. . . .• .. . , . , ,- - - -,

,- . , : . o - . . . . . . . . .

"2m" "'""1 X~2S 2 1-"et = 14 5 f()-- - /k n t ds (A-6)

.-. ,)

Equation (A-6) is in a form which begins to resemble

the result obtained by Meyer (Meyer, unpublished). The final

expression for the dielectric function will be the same as

*. Meyer's, except that scaled arguments will be used.

The reduced Fermi energy, ne Ve/kbT , is introduced

here. One defines

1 (2m*k T) (A-7)

2 2and makes the change of variables x = (IS/2mekbT)s . Then

" equation (A-6) becomes

geme* f) f 8 oon-- dx (A-8)

x

Now, equation (A-8) does not contain the electron-hole

pair density, n. To introduce the density, it is necessary

to evaluate

n = d 3k f[(Ee(k) - Ue)/kbT (A-9)

When the integration in equation (A-9) is performed, and the

_result is substituted into equation (A-8), equations (20)

through (23) for the dielectric function are obtained.

45

VITA

. •Barry Scott Davies was born on 6 September 1951 in New

Haven Connecticut. He graduated from high school in Daubury

Connecticut in 1970 and attended the Paier School of Art in

Hamden Ct, where he studied technical illustration, for one

year. He then attended Colby College in Waterville Maine

from which he received his Bachelor of Arts degree in

physics in 1975. After two years as laborer, he taught

physics at Worester Academy in Worcester Massachusetts for

one year, and then attended graduate school at the Univer-

sity of Massachusetts (Amherst) antil January of 1981. He

entered the Air Force Officer Training School in February

of 1981 and was assigned to AFIT upon graduation.

46

SECUmITY CLASSIFICATION OF THIS PAGE (When Does Entered) U PC tYs 1f .

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM

1APOT NUMERtP/ - 12. GOV S 4 ACC ECIP lENT'S CATALOG NUMBER

14. TITLE (and Subrtile) I. TYPE OF REPORT & PERIOD COVERED

EFFECTS OF CONDUCTION BAND ANISOTROPY MS ThesisON THE EXCITON-PLASMA MOTT TRANSITION

IN INDIRECT GAP SEMICONDUCTORS G. PERFORMING ORG. REPORT NUMBER

7. AUTNOR(.) S. CONTRACT OR GRANT NUMBER(&)

Barry S. Davies2LT

S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKI AREA & WORK UNIT NUMBERS

Air Force Institute of Technology (AFIT-EN) 61102FWright-Patterson AFB, Ohio 45433 2306 Ri 01

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

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t)ICLASSI Fi-DSo. DECLASSI FI CATION/DOWNGRADING

SCHEDULE

UL DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abetract entered in Block 20. ii different from Report)

Approved for public release; distribution unlimited

I4. SUPPLEMENTARY NOTES

x~roe I AW ATE 19017.

=" x. 9 J AN "' 3Ar Force jr .n~ tu te oj jechn,iogy PI'Wg1ghtg.Pailtaon AF4 OM 4

19. KEY WORDS (Continue on reveree aide if necessary mnd identify by ark number)

Mott Transition, Exciton Ionization,Random Phase Approximation

20. ABSTRACT (Continue on reverse Ode It necesearey o Identify by bleak mber)

A theory is developed for the incorporation of conduc-tion band anisotropy into the analysis of the exciton-plasmaMott transition in indirect gap semiconductors. Ellipsoidalenergy surfaces are assumed for the electrons while spheri-cal energy surfaces are retained for holes. Static electron-hole screening in the random phase approximation is assumed.

". ,,SFORM173 D

DJ FAN 1473 EDITIONOF INOVsis OSOLETE , u cL4%s I f lij- SECURITY CLASSIFICATION OF THIS PAGE (W0gen Date Entered)

SECURITY CLASSIF'CATION or THIS PAGIE(Wht. Dee. Inhered)

The Mott transition is associated with the electron-hole pairdensity at which the exciton binding energy in the assumedpotential is zero. The binding energy is computed variationally.

It is found that the electron anisotropy causes the Motttransition to shift to higher densities. It is also found that,in the absence of screening, the exciton binding energy is notsignificantly affected by the electron anisotropy. It is thusconcluded that the shift to higher densities is due largelyto the reduced ability of anisotropic electrons to screen.

- , .. . . " ; -

3 A - 0 A;

-- .: - -oC.°

--'

7. 7

, .= ,. . -, , .. . . "- - . , . , : .

'-4..

. . ",- " .

5 ~J SCCURIY CLASSIFICATION OF THIS PAGcEtIrhen Date Enetd)

0~ I-